Exact Controllability and Feedback Stabilization from a Boundary for the Navier-Stokes Equations

Abstract

For 2D Navier-Stokes equations defined in a bounded domain Ω we study stabilization of solution near a given steady-state flow $\hat v(x)$ by means of feedback control defined on a part Γ of boundary ∂Ω. New mathematical formalization of feedback notion is proposed. With its help for a prescribed number σ > 0 and for an initial condition v ₀(x) placed in a small neighbourhood of $\hat v(x)$ a control u(t, x′), x ∈ Γ, is constructed such that solution v(t, x) of obtained boundary value problem for 2D Navier-Stokes equations satisfies the inequality:

$$ \left\| {v(t, \cdot ) - \hat v} \right\|_{H^1 } \leqslant ce^{ - \sigma t} for t \geqslant 0. $$

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